# Applications and motivation of abstract linear algebra topics for engineers

This semester I'm teaching introductory linear algebra for engineering students, and I don't think I'm doing a good job explaining why these topics are important; specifically, everything having to do with linear transformations. I'm a physicist (physics student, really), and so when thinking about linear algebra the first thing that comes to mind is eigenvectors, which are not covered in this course.

This week I have to teach some of the most abstract stuff yet: Kernels of linear transformations, images and preimages (and showing that they are subspaces of $\mathbb{R}^n$), defining a linear transformation by its action on a basis, the rank-nullity theorem, composing linear transformations, etc. Personally I love this subject, because I think that once you really understand it, everything fits together in the most beautiful way and all the connections between the theorems seem obvious; sadly, this isn't very useful for my students who are still getting used to multiplying matrices.

What are some examples I can give to show why this stuff is useful in engineering? I'm particularly interested in the subjects I mentioned above. Please note that not teaching them is not an option: It's not up to me to decide what is covered and what isn't, and I don't write the exams.

• One suggestion is to glance through some upper level engineering texts at your university's bookstore. You'll see matrices (and possibly linear functions) used in dynamics texts (chapter on vibrations), for instance, although eigenvalues (which you say are not covered) wind up playing a major role. Maybe look in some fluid and continuum mechanics texts, although since you say you're a physics student, I presume you already know the applications there. – Dave L Renfro Oct 28 '15 at 18:30
• Have you checked through the textbook Matrix Analysis and Applied Linear Algebra (pdf; 890p)? You could search through that text for various strings, e.g,. appli, and see which examples would be worth mentioning. – Benjamin Dickman Oct 28 '15 at 23:39
• I would also add that robotics uses transformation matrices quite a bit and LOTS of compositions!! I remember helping my now-husband with that when he took a robotics class. This uses "action on a basis" quite a bit and definitely involves images and preimages. I should look for my husband's robotics text for references. Matrices should also help with Kirschoff's rules for circuits. When do you have enough information to solve a circuit? This would have to do with matrix rank. Lastly network analysis also uses matrices but requires some graph theory. Not an engineer, so just commenting. – Opal E Nov 30 '15 at 22:01

## 4 Answers

Of course it depends on how much time you're willing to spend on this. If the answer is "very little" then no chance that you can say something more than "in the future this will be useful for you"...

So I start from the idea that you can devote at least one hour to this. And it better be a well prepared one, otherwise time will not be enough.

Codes: I do not think that it is difficult to explain as Joseph's says, at least if you do not really expect to arrive to the point of error-correcting ones. The idea of check digit in a linear code is exactly equivalent to the kernel of a linear transformation. If they have a book with them they have an ISBN code written on it and you can have then check that the last digit satisfies a linear equation in the vector space $\mathbb Z_{11}^{11}$: https://en.wikipedia.org/wiki/International_Standard_Book_Number Similarly for bar-codes.

Linear optimization: you can show how to use matrices and linear equations to solve an easy optimization problem, e.g. the "diet problem" with restrictions. There is no need to develop a full theory. Just build up an example with resources and consumption of resources, and formalize it with a system of linear equations. Look at some examples here. You will have your way to decide whether you want to prove inequlities or only equalities and can easily adapt them... http://www.shelovesmath.com/algebra/advanced-algebra/linear-programming/ (BTW if you students are addicted to some MORPG you can use a word problem with resources appearing in that game)

Least squares method. It is an approximation problem turned into a linear algebra problem. Not too difficult to explain (you better have had some discussions about norms, though) and easy to see that it can lead to linear systems with a huge number of equations: good place to remind them why Gauss reduction is much more effective than algorithms based on determinants...

Markov chains in disguise: though you do not have eigenvalues and eigenvectors you can show that matrix powers have a role in understanding the behaviour of some linear dynamical system. The examples I do prefer are:

• dynamic of a population: divide a population in young, adult, old. Define a fertility number and a dying rate to each of this and see how the population evolves starting from a fixed initial vector distribution. This is computing powers of a 3x3 matrix.
• reallocation of resources: say you're car renting in NY-Chicago-LA and have a monthly rate of cars being returned at the same rental or to another in the country. Starting from any distribution of cars at time 0 determine the distribution of cars after a fixed amount of time.

Two Four ideas:

(1) "composing linear transformations": Use rotation, scaling, and shearing. If you extend to homogenous coordinates, you can include translations. Fundamental to all computer graphics. Explore which combinations of these transformations always commute, and which sometimes do not commute. In $\mathbb{R}^2$ and in $\mathbb{R}^3$. E.g., rotations in $\mathbb{R}^3$ in general do not commute.

(2) "the rank-nullity theorem": Derive the consequence that there is no injective (1-to-1) linear transformation from $\mathbb{R}^3 \to \mathbb{R}^2$. And the reverse, that there is no surjective (onto) linear transformation from $\mathbb{R}^2 \to \mathbb{R}^3$. These are quite natural conclusions.

(3) "Kernels of linear transformations": They play a role in error-correcting codes, but that would not be easy to explain. One intuitive example is: the kernel of projection onto a plane is the set of vectors orthogonal to the plane.

(4) "its action on a basis": To understand reflection of a ray in a mirror, it is easier to use basis vectors tangent to and perpendicular to the plane of the mirror, rather than a standard basis.

• +1 for connection with comp graphics. absolutely essential. – celeriko Oct 28 '15 at 0:48

Engineers at our school are not required to take the full Linear Algebra course. However, they take a 1-credit course called "Linear Algebra for Differential Equations." So I think the answer might be to look into some Differential Equations material where the entries of the matrices, the basis elements, and the eigenvectors are all functions (usually with understandable conceptual meanings).

My father, a retired engineering professor, told me that engineers think largely in terms of systems of equations. The theory is important, but only to the extent that it serves that end.

For instance, students will want to know how and why a system of equations "span" a particular space. As for solution methods, the foundation method is nowadays called "lower echelon row reduction" (my father used a different term 50 years ago), using linear transformations. The ideas of kernels and images really have to do with which equations (and co-efficients) are left in the system after the reduction process. That's what engineers are most interested in.